1. Engineering Economic Analysis
Chapter 4: Part 3
Other Interest Formulas
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

2. Nominal and Effective Interest
Nominal interest rate/year: the annual interest rate w/o considering the effect of any compounding.
12%/year
Interest rate/period: the nominal interest rate/year divided by the number of interest compounding periods.
12%/year/12 months/year = 1%/period
Effective interest rate/year: the annual interest rate taking into account the effect of the compounding periods in the year.
12%/year compounded monthly is equivalent to 12.68%/year compounded yearly
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

3. Nominal and Effective Interest
Assume i = 5%, compounded semi-annually. How much interest accrues on $100 over 1 year at this rate?
Interest accrued = $100 · (F/P, i, n) - $100
Here
n = 2
We get $100 · ( )2 – $100 = $100 · (1.051) - $100 = $ $100 = $5.10
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

4. Nominal and Effective Interest
What interest rate, compounded annually results in F = $105.10, when P = 100 over 1 year
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

5. Nominal and Effective Interest
How does 5.1% per year relate to 5% compounded semiannually?
5% is the nominal interest rate
5.1% is the effective interest rate
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

6. Nominal and Effective Interest
r is the nominal interest rate per interest period. It is the rate not considering compounding
i is the effective interest rate per interest period. It is the rate considering compounding
ia is the effective interest rate per year
m number of compounding periods per time period
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

7. Nominal and Effective Interest
The effective interest rate ia is defined to be:
This is why 5% compounded semiannually means that ia is 5.1% per year
m = 2, r = 5% and i = 2.5%
In Excel: EFFECT(nom, nper)
nom = the nominal rate for an entire year, r
nper = the number of compoundings in a year, m
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

8. Example 1
You go to a check cashing store to get a $100 advance on your pay check. They are willing to do so provided you write a check for $120 dated for 1 week from now.
What nominal interest rate per year are you paying?
What effective interest rate per year are you paying?
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

9. Example 1 What nominal interest rate per year are you paying?
First determine the weekly interest.
Remember nominal means no interest
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

10. Example 1 What effective interest rate per year are you paying?
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

11. Example 1
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

12. Example 2
A bank pays 8% nominal interest per year, compounded quarterly. A person deposits $5,000 at time 0. After the end of each of the next 5 years she wants to withdraw an equal amount. What is this amount?
If compounding interval doesn’t equal cash flow period, we need to be sure to notice
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

13. Example 2
This is not a uniform series problem because we don’t have a cash flow each time, but interest is compounded each time. There are at least two ways to solve this.
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

14. Example 2
Find
Then
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

15. Example 2
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

16. Example 2
Translate 8% nominal interest compounded quarterly into an effective annual rate so we have 5 one year periods.
r = 8%
m = 4
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

17. Example 2
We now know that we have an effective annual interest of 8.24% and now we can use the following cash flow diagram.
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

18. Example 2
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

19. Example 2
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

20. Continuous Compounding
As m, the number of compounding subperiods per year grows to ∞, the interest compounds continuously.
m↑ and r/m ↓
Under continuous compounding, the effective interest rate per year is:
In Excel, just use the exponential directly
Many credit cards use continuous compounding
Where e is the exponential function
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

21. Single Payment – Continuous Compounding
When we have single payments with continuous compounding we replace the (1+i) with e and n with r · n
So we see that the formulas are very similar. The bracket instead of the parenthesis for the functional notation imply continuous compounding. For these we have to use a different table.
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

22. Example 3
What is the amount of interest earned on $2000 for 2 years earning 5% nominal interest compounded continuously?
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

23. Example 3
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

24. Example 4
How long will it take $2,000 to double to $4,000 at 10% nominal interest compounded continuously?
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

25. Example 4 This is a perfect application of Goal Seek
Use a value of 1 to make sure the formula in B9 works
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

26. Example 5
What is the effective annual interest rate of 6% compounded continuously?
With Excel, you can see how the non-continuous compounding formula performs with more and more compounding periods:
Do the differences in these values matter? Probably, if you are Bank of America getting mortgage payments, yes.
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

27. Uniform Series – Continuous Compounding at Nominal Rate r per Period
Just use i = er – 1 in the uniform series equations:
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

28. Example 6
How much money will accrue in an account earning 5% compounded continuously for 5 years if $500 is deposited each year?
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

29. Money flows continuously over one period, like:
Continuous Uniform Cash Flow (1 Period) with Continuous Compounding at rate r
Money flows continuously over one period, like:
Money out of an ATM
Money into a credit card company in the form of payments
Let the continuous, uniform cash flow totaling be distributed over m subperiods within one
period (n = 1). Thus is the cash flow at the
end of each subperiod
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

30. Continuous Uniform Cash Flow (1 Period) with Continuous Compounding at rate r
We use the following formula to find the future value n periods in the future.
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

31. Continuous Uniform Cash Flow (1 Period) with Continuous Compounding at rate r
We use the following formula to find the present of a continuous payment n periods in the future.
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

32. Example 7
A gas station ATM takes payments. The gas station receives $40,000 per month, credited continuously to their account during each month. If they earn 9% nominal interest compounded continuously, how much money will the bank account have, if they start at 0 at time 0, after one month?
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

33. Example 7
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.

34. Example 7 No special tricks for Excel
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.